Hierarchical Edge Bundling

Constructing Hierarchical Edge Bundling charts for molecular data.

Molecular analytes

Female data

Load required libraries, data and create the dendrogram

This code loads necessary R packages, loads the female data, standardizes the data to one unit variance across sampled groups, calculates euclidean distances, performs hierarchical clustering and creates a dendrogram.

Code

# Load required libraries
library(readxl)
library(dplyr)
library(stats)
library(ggplot2)
library(ggdendro)
library(igraph)
library(ggraph)
library(tidygraph)

# Step 1: Load and standardize the data
data <- read_excel(here::here("Data/Mol_raw_data.xlsx"), sheet = 1)
data <- as.data.frame(data)

# Ensure the first column becomes the row names
rownames(data) <- data[,1]
data <- data[,-1]

#Standardise the data 
scaled_data <- vegan::decostand(data, method="standardize", MARGIN = 1)

# Step 2: Calculate Euclidean distances
dist_matrix <- dist(scaled_data, method = "euclidean")

# Step 3: Apply hierarchical clustering
hc <- hclust(dist_matrix, method = "ward.D2")

# Step 4 : Convert to dendrogram 
dendro <- as.dendrogram(hc)

Create dataframes

This code creates three dataframes necessary for plotting the charts: edges, vertices and connections

Code

#Step 5: Use ggraph 
hierarchy_graph = as_tbl_graph(dendro)

#Hierarchal dataframe 
edge_df <- hierarchy_graph %>%
  activate(edges) %>%
  as_tibble() %>%
  rename(from = from, to = to)

# Extract vertices dataframe
vertex_df <- hierarchy_graph %>%
  activate(nodes) %>%
  as_tibble() %>%
  mutate(node_id = row_number())

vertices_my <- vertex_df %>%
  mutate(id = row_number(),
         name = label,
         shortName = label,  # Assuming you don't have a shorter name
         group = ifelse(leaf, "leaf", "internal")) %>%
  select(id, name, shortName, group, leaf, height, members) %>%
  arrange(name) %>%
  mutate(name = factor(name, levels = unique(name)))

# Check for NA or empty names and replace them with a placeholder
vertices_my <- vertices_my %>%
  mutate(shortName = ifelse(is.na(shortName) | shortName == "", paste0("Node", row_number()), shortName)) %>%
  mutate(name = ifelse(is.na(name) | name == "", paste0("Node_", row_number()), name))

# Create connections_my dataframe
connections_my_df <- edge_df %>%
  left_join(vertex_df, by = c("from" = "node_id")) %>%
  left_join(vertex_df, by = c("to" = "node_id"), suffix = c("_from", "_to"))

connections_my <- connections_my_df %>%
  select(from, to) %>% #This selects only the 'from' and 'to' columns from the original dataframe.
  mutate(from = vertices_my$name[from],
         to = vertices_my$name[to]) %>% #This replaces the values in the 'from' and 'to' columns with corresponding names from the vertices_my dataframe.
  distinct() #This removes any duplicate rows from the resulting dataframe.

connections_my <- connections_my %>%
  filter(from != "" & to != "") #This keeps only the rows where from and to are not empty strings 

# Preparation to draw labels properly:
# Extract vertex data
vertices_my <- hierarchy_graph %>%
  activate(nodes) %>%
  as_tibble() %>%
  mutate(
    id = row_number(),
    name = label,
    shortName = label,  # Keep original labels
    group = ifelse(leaf, "leaf", "internal"),
    unique_name = make.unique(as.character(name), sep = "_")
  ) %>%
  select(id, name, shortName, unique_name, group, leaf, height, members)

# Correct any empty or NA names only for internal nodes
vertices_my <- vertices_my %>%
  mutate(
    name = ifelse(!leaf & (is.na(name) | name == ""), paste0("Node_", row_number()), name),
    shortName = ifelse(!leaf & (is.na(shortName) | shortName == ""), paste0("Node", row_number()), shortName)
  )

# Prepare label angles for leaves
leaf_vertices <- vertices_my %>% 
  filter(leaf) %>%
  mutate(
    leaf_id = row_number(),
    angle = 90 - 360 * (leaf_id - 1) / n(),
    hjust = ifelse(angle < -90, 1, 0),
    angle = ifelse(angle < -90, angle + 180, angle)
  )

# Update vertices_my with leaf angles
vertices_my <- vertices_my %>%
  left_join(leaf_vertices %>% select(id, angle, hjust), by = "id")

The circular dendrogram

This code plots a circular dendrogram from the hierarchical clustering results

Code

#Create the graph object 
mygraph <- graph_from_data_frame(edge_df, vertices = vertices_my)

# Basic dendrogram ----
ggraph(mygraph, layout = 'dendrogram', circular = TRUE) +
  geom_edge_link(size = 0.4, alpha = 0.1) +
  geom_node_text(aes(x = x*1.01, y = y*1.01, filter = leaf, label = shortName, angle = angle, hjust = hjust), size = 1.5, alpha = 1) +
  coord_fixed() +
  theme_void() +
  theme(
    legend.position = "none",
    plot.margin = unit(c(0,0,0,0),"cm"),
  ) +
  expand_limits(x = c(-1.2, 1.2), y = c(-1.2, 1.2))

The hierarchical edge bundling

This code plots a hierarchical edge bundling chart from the hierarchy defined in the dendrogram

Code

#Correct the connections_my dataframe with valid connections 
# Get leaf nodes
leaf_nodes <- V(mygraph)$name[V(mygraph)$leaf]

# Create all possible combinations of leaf nodes
all_combinations <- expand.grid(from = leaf_nodes, to = leaf_nodes) %>%
  mutate(
    from = as.character(from),
    to = as.character(to)
  )

# Remove self-connections and duplicate connections
connections_my_corrected <- all_combinations %>%
  filter(from != to) %>%  # Remove self-connections
  mutate(
    pair = pmin(from, to),
    pair_to = pmax(from, to)
  ) %>%
  distinct(pair, pair_to) %>%  # Remove duplicates
  select(from = pair, to = pair_to)

# Convert node names to indices
from <- match(connections_my_corrected$from, V(mygraph)$name)
to <- match(connections_my_corrected$to, V(mygraph)$name)

# Plot the hierarchical edge bundle 
ggraph(mygraph, layout = 'dendrogram', circular = TRUE) +
  geom_conn_bundle(data = get_con(from = from, to = to), 
                   alpha = 0.2,  # Reduced alpha for better visibility
                 #  colour="#69b3a2", 
                   tension = 1.5,
  aes(colour=after_stat(index))) +
  scale_edge_colour_distiller(palette = "RdPu") +
  geom_node_point(aes(filter = leaf), size = 2, color = "black") +
  geom_node_text(aes(x = x*1.05, y=y*1.05, filter = leaf, label=shortName, angle = angle, hjust=hjust), 
                 size=2, alpha=1) +
  coord_fixed() +
  theme_void() +
  theme(
    legend.position="none",
    plot.margin=unit(c(0,0,0,0),"cm"),
  ) +
  expand_limits(x = c(-1.3, 1.3), y = c(-1.3, 1.3))

Nbclust

The NbClust function is being used to determine the optimal number of clusters in our data. It applies multiple clustering validity indices and methods to suggest the best number of clusters, providing a consensus view based on various criteria to help make a more robust decision about the appropriate cluster count for our dataset.

Code

library(stringr)

library(NbClust)
methods <- c("ward.D2")

for (m in methods) {
  # Capture all output
  output <- capture.output({
    result <- NbClust(data = scaled_data, 
                      distance = "euclidean", 
                      min.nc = 2, 
                      max.nc = 10, 
                      method = m, 
                      index = "all")
  })
  
  # Join the output into a single string
  output_text <- paste(output, collapse = "\n")
  
  # Split the output by the asterisk separator
  split_output <- str_split(output_text, "\\*{10,}")[[1]]
  
  # Check if we have at least 3 parts
  if (length(split_output) >= 3) {
    # Trim whitespace from the second part
    second_part <- str_trim(split_output[2])
    
    # Split the second part into lines
    lines <- str_split(second_part, "\n")[[1]]
    
    # Format the output
    formatted_output <- character(0)
    for (line in lines) {
      if (str_detect(line, "Conclusion")) {
        formatted_output <- c(formatted_output, "", "Conclusion", "")
      } else if (str_detect(line, "^\\*")) {
        formatted_output <- c(formatted_output, line)
      }
    }
    
    # Join the formatted lines
    formatted_output <- paste(formatted_output, collapse = "\n")
    
    # Print the results
    cat(paste("Results for method:", m, "\n\n"))
    cat(formatted_output)
    cat("\n\n")  # Add some space between results for different methods
  } else {
    cat(paste("Unexpected output format for method:", m, "\n"))
  }
}

Results for method: ward.D2

• Among all indices:
  
• 3 proposed 2 as the best number of clusters
• 11 proposed 3 as the best number of clusters
• 2 proposed 5 as the best number of clusters
• 7 proposed 10 as the best number of clusters

Conclusion

• According to the majority rule, the best number of clusters is 3

Plot the clusters

Code

cluster_groups <- cutree(hc, k = 3)

# Add cluster information to vertices_my
vertices_my <- vertices_my %>%
  mutate(cluster = factor(cluster_groups[match(name, names(cluster_groups))]))

library(RColorBrewer)

# Select three colors from the "Paired" palette
cluster_colors <- brewer.pal(3, "Paired")

#Create the graph object 
mygraph <- graph_from_data_frame(edge_df, vertices = vertices_my)

#Correct the connections_my dataframe with valid connections 
# Get leaf nodes
leaf_nodes <- V(mygraph)$name[V(mygraph)$leaf]

# Create all possible combinations of leaf nodes
all_combinations <- expand.grid(from = leaf_nodes, to = leaf_nodes) %>%
  mutate(
    from = as.character(from),
    to = as.character(to)
  )

# Remove self-connections and duplicate connections
connections_my_corrected <- all_combinations %>%
  filter(from != to) %>%  # Remove self-connections
  mutate(
    pair = pmin(from, to),
    pair_to = pmax(from, to)
  ) %>%
  distinct(pair, pair_to) %>%  # Remove duplicates
  select(from = pair, to = pair_to)

# Convert node names to indices
from <- match(connections_my_corrected$from, V(mygraph)$name)
to <- match(connections_my_corrected$to, V(mygraph)$name)

# Plot the hierarchical edge bundle with colored points
ggraph(mygraph, layout = 'dendrogram', circular = TRUE) +
  geom_conn_bundle(data = get_con(from = from, to = to), 
                   tension = 1.5,
                   aes(colour=after_stat(index))) +
  scale_edge_colour_distiller(palette = "RdPu") +
  geom_node_point(aes(filter = leaf, color = cluster),  # Move points slightly outward
                  size = 2, alpha = 1) +
  geom_node_text(aes(x = x * 1.05, y = y * 1.05,  # Move text even further outward
                     filter = leaf, label = shortName, 
                     angle = angle, hjust = hjust), 
                 size = 2, alpha = 1) +
  scale_color_manual(values = cluster_colors) +
  coord_fixed() +
  theme_void() +
  theme(
    legend.position = "none",
    plot.margin = unit(c(0,0,0,0), "cm"),
  ) +
  expand_limits(x = c(-1.3, 1.3), y = c(-1.3, 1.3))  # Further expanded limits to accommodate text and points

Male data

Load data and create the dendrogram

This code loads the male data, standardizes the data to one unit variance across sampled groups, calculates euclidean distances, performs hierarchical clustering and creates a dendrogram.

Code

# Step 1: Load and standardize the data
data <- read_excel(here::here("Data/Mol_raw_data.xlsx"), sheet = 2)
data <- as.data.frame(data)

# Ensure the first column becomes the row names
rownames(data) <- data[,1]
data <- data[,-1]

#Standardise the data 
scaled_data <- vegan::decostand(data, method="standardize", MARGIN = 1)

# Step 2: Calculate Euclidean distances
dist_matrix <- dist(scaled_data, method = "euclidean")

# Step 3: Apply hierarchical clustering
hc <- hclust(dist_matrix, method = "ward.D2")

# Step 4 : Convert to dendrogram 
dendro <- as.dendrogram(hc)

Create dataframes

This code creates three dataframes necessary for plotting the charts: edges, vertices and connections

Code

#Step 5: Use ggraph 
hierarchy_graph = as_tbl_graph(dendro)

#Hierarchal dataframe 
edge_df <- hierarchy_graph %>%
  activate(edges) %>%
  as_tibble() %>%
  rename(from = from, to = to)

# Extract vertices dataframe
vertex_df <- hierarchy_graph %>%
  activate(nodes) %>%
  as_tibble() %>%
  mutate(node_id = row_number())

vertices_my <- vertex_df %>%
  mutate(id = row_number(),
         name = label,
         shortName = label,  # Assuming you don't have a shorter name
         group = ifelse(leaf, "leaf", "internal")) %>%
  select(id, name, shortName, group, leaf, height, members) %>%
  arrange(name) %>%
  mutate(name = factor(name, levels = unique(name)))

# Check for NA or empty names and replace them with a placeholder
vertices_my <- vertices_my %>%
  mutate(shortName = ifelse(is.na(shortName) | shortName == "", paste0("Node", row_number()), shortName)) %>%
  mutate(name = ifelse(is.na(name) | name == "", paste0("Node_", row_number()), name))

# Create connections_my dataframe
connections_my_df <- edge_df %>%
  left_join(vertex_df, by = c("from" = "node_id")) %>%
  left_join(vertex_df, by = c("to" = "node_id"), suffix = c("_from", "_to"))

connections_my <- connections_my_df %>%
  select(from, to) %>% #This selects only the 'from' and 'to' columns from the original dataframe.
  mutate(from = vertices_my$name[from],
         to = vertices_my$name[to]) %>% #This replaces the values in the 'from' and 'to' columns with corresponding names from the vertices_my dataframe.
  distinct() #This removes any duplicate rows from the resulting dataframe.

connections_my <- connections_my %>%
  filter(from != "" & to != "") #This keeps only the rows where from and to are not empty strings 

# Preparation to draw labels properly:
# Extract vertex data
vertices_my <- hierarchy_graph %>%
  activate(nodes) %>%
  as_tibble() %>%
  mutate(
    id = row_number(),
    name = label,
    shortName = label,  # Keep original labels
    group = ifelse(leaf, "leaf", "internal"),
    unique_name = make.unique(as.character(name), sep = "_")
  ) %>%
  select(id, name, shortName, unique_name, group, leaf, height, members)

# Correct any empty or NA names only for internal nodes
vertices_my <- vertices_my %>%
  mutate(
    name = ifelse(!leaf & (is.na(name) | name == ""), paste0("Node_", row_number()), name),
    shortName = ifelse(!leaf & (is.na(shortName) | shortName == ""), paste0("Node", row_number()), shortName)
  )

# Prepare label angles for leaves
leaf_vertices <- vertices_my %>% 
  filter(leaf) %>%
  mutate(
    leaf_id = row_number(),
    angle = 90 - 360 * (leaf_id - 1) / n(),
    hjust = ifelse(angle < -90, 1, 0),
    angle = ifelse(angle < -90, angle + 180, angle)
  )

# Update vertices_my with leaf angles
vertices_my <- vertices_my %>%
  left_join(leaf_vertices %>% select(id, angle, hjust), by = "id")

The circular dendrogram

This code plots a circular dendrogram from the hierarchical clustering results

Code

#Create the graph object 
mygraph <- graph_from_data_frame(edge_df, vertices = vertices_my)

# Basic dendrogram ----
ggraph(mygraph, layout = 'dendrogram', circular = TRUE) +
  geom_edge_link(size = 0.4, alpha = 0.1) +
  geom_node_text(aes(x = x*1.01, y = y*1.01, filter = leaf, label = shortName, angle = angle, hjust = hjust), size = 1.5, alpha = 1) +
  coord_fixed() +
  theme_void() +
  theme(
    legend.position = "none",
    plot.margin = unit(c(0,0,0,0),"cm"),
  ) +
  expand_limits(x = c(-1.2, 1.2), y = c(-1.2, 1.2))

The hierarchical edge bundling

This code plots a hierarchical edge bundling chart from the hierarchy defined in the dendrogram

Code

#Correct the connections_my dataframe with valid connections 
# Get leaf nodes
leaf_nodes <- V(mygraph)$name[V(mygraph)$leaf]

# Create all possible combinations of leaf nodes
all_combinations <- expand.grid(from = leaf_nodes, to = leaf_nodes) %>%
  mutate(
    from = as.character(from),
    to = as.character(to)
  )

# Remove self-connections and duplicate connections
connections_my_corrected <- all_combinations %>%
  filter(from != to) %>%  # Remove self-connections
  mutate(
    pair = pmin(from, to),
    pair_to = pmax(from, to)
  ) %>%
  distinct(pair, pair_to) %>%  # Remove duplicates
  select(from = pair, to = pair_to)

# Convert node names to indices
from <- match(connections_my_corrected$from, V(mygraph)$name)
to <- match(connections_my_corrected$to, V(mygraph)$name)

# Plot the hierarchical edge bundle 
ggraph(mygraph, layout = 'dendrogram', circular = TRUE) +
  geom_conn_bundle(data = get_con(from = from, to = to), 
                   alpha = 0.2,  # Reduced alpha for better visibility
                 #  colour="#69b3a2", 
                   tension = 1.5,
  aes(colour=after_stat(index))) +
  scale_edge_colour_distiller(palette = "BuPu") +
  geom_node_point(aes(filter = leaf), size = 2, color = "black") +
  geom_node_text(aes(x = x*1.05, y=y*1.05, filter = leaf, label=shortName, angle = angle, hjust=hjust), 
                 size=2, alpha=1) +
  coord_fixed() +
  theme_void() +
  theme(
    legend.position="none",
    plot.margin=unit(c(0,0,0,0),"cm"),
  ) +
  expand_limits(x = c(-1.3, 1.3), y = c(-1.3, 1.3))

Nbclust

The NbClust function is being used to determine the optimal number of clusters in our data. It applies multiple clustering validity indices and methods to suggest the best number of clusters, providing a consensus view based on various criteria to help make a more robust decision about the appropriate cluster count for our dataset.

Code

library(stringr)

library(NbClust)
methods <- c("ward.D2")

for (m in methods) {
  # Capture all output
  output <- capture.output({
    result <- NbClust(data = scaled_data, 
                      distance = "euclidean", 
                      min.nc = 2, 
                      max.nc = 10, 
                      method = m, 
                      index = "all")
  })
  
  # Join the output into a single string
  output_text <- paste(output, collapse = "\n")
  
  # Split the output by the asterisk separator
  split_output <- str_split(output_text, "\\*{10,}")[[1]]
  
  # Check if we have at least 3 parts
  if (length(split_output) >= 3) {
    # Trim whitespace from the second part
    second_part <- str_trim(split_output[2])
    
    # Split the second part into lines
    lines <- str_split(second_part, "\n")[[1]]
    
    # Format the output
    formatted_output <- character(0)
    for (line in lines) {
      if (str_detect(line, "Conclusion")) {
        formatted_output <- c(formatted_output, "", "Conclusion", "")
      } else if (str_detect(line, "^\\*")) {
        formatted_output <- c(formatted_output, line)
      }
    }
    
    # Join the formatted lines
    formatted_output <- paste(formatted_output, collapse = "\n")
    
    # Print the results
    cat(paste("Results for method:", m, "\n\n"))
    cat(formatted_output)
    cat("\n\n")  # Add some space between results for different methods
  } else {
    cat(paste("Unexpected output format for method:", m, "\n"))
  }
}

Results for method: ward.D2

• Among all indices:
  
• 8 proposed 2 as the best number of clusters
• 6 proposed 3 as the best number of clusters
• 1 proposed 4 as the best number of clusters
• 2 proposed 5 as the best number of clusters
• 7 proposed 10 as the best number of clusters

Conclusion

• According to the majority rule, the best number of clusters is 2

Plot the clusters

Code

cluster_groups <- cutree(hc, k = 2)

# Add cluster information to vertices_my
vertices_my <- vertices_my %>%
  mutate(cluster = factor(cluster_groups[match(name, names(cluster_groups))]))

library(RColorBrewer)

# Select three colors from the "Paired" palette
cluster_colors <- brewer.pal(2, "Paired")

#Create the graph object 
mygraph <- graph_from_data_frame(edge_df, vertices = vertices_my)

#Correct the connections_my dataframe with valid connections 
# Get leaf nodes
leaf_nodes <- V(mygraph)$name[V(mygraph)$leaf]

# Create all possible combinations of leaf nodes
all_combinations <- expand.grid(from = leaf_nodes, to = leaf_nodes) %>%
  mutate(
    from = as.character(from),
    to = as.character(to)
  )

# Remove self-connections and duplicate connections
connections_my_corrected <- all_combinations %>%
  filter(from != to) %>%  # Remove self-connections
  mutate(
    pair = pmin(from, to),
    pair_to = pmax(from, to)
  ) %>%
  distinct(pair, pair_to) %>%  # Remove duplicates
  select(from = pair, to = pair_to)

# Convert node names to indices
from <- match(connections_my_corrected$from, V(mygraph)$name)
to <- match(connections_my_corrected$to, V(mygraph)$name)

# Plot the hierarchical edge bundle with colored points
ggraph(mygraph, layout = 'dendrogram', circular = TRUE) +
  geom_conn_bundle(data = get_con(from = from, to = to), 
                   tension = 1.5,
                   aes(colour=after_stat(index))) +
  scale_edge_colour_distiller(palette = "BuPu") +
  geom_node_point(aes(filter = leaf, color = cluster, 
                      x = x, y = y),  # Move points slightly outward
                  size = 2, alpha = 1) +
  geom_node_text(aes(x = x * 1.05, y = y * 1.05,  # Move text even further outward
                     filter = leaf, label = shortName, 
                     angle = angle, hjust = hjust), 
                 size = 2, alpha = 1) +
  scale_color_manual(values = cluster_colors) +
  coord_fixed() +
  theme_void() +
  theme(
    legend.position = "none",
    plot.margin = unit(c(0,0,0,0), "cm"),
  ) +
  expand_limits(x = c(-1.3, 1.3), y = c(-1.3, 1.3))  # Further expanded limits to accommodate text and points

Molecular analytes and cardiometabolic health markers

Female data

Load data and create the dendrogram

This code loads the female data, standardizes the data to one unit variance across sampled groups, calculates euclidean distances, performs hierarchical clustering and creates a dendrogram.

Code

# Step 1: Load and standardize the data
data <- read_excel(here::here("Data/Mol_raw_data.xlsx"), sheet = 3)
data <- as.data.frame(data)

# Ensure the first column becomes the row names
rownames(data) <- data[,1]
data <- data[,-1]

#Standardise the data 
scaled_data <- vegan::decostand(data, method="standardize", MARGIN = 1)

# Step 2: Calculate Euclidean distances
dist_matrix <- dist(scaled_data, method = "euclidean")

# Step 3: Apply hierarchical clustering
hc <- hclust(dist_matrix, method = "ward.D2")

# Step 4 : Convert to dendrogram 
dendro <- as.dendrogram(hc)

Create dataframes

This code creates three dataframes necessary for plotting the charts: edges, vertices and connections

Code

#Step 5: Use ggraph 
hierarchy_graph = as_tbl_graph(dendro)

#Hierarchal dataframe 
edge_df <- hierarchy_graph %>%
  activate(edges) %>%
  as_tibble() %>%
  rename(from = from, to = to)

# Extract vertices dataframe
vertex_df <- hierarchy_graph %>%
  activate(nodes) %>%
  as_tibble() %>%
  mutate(node_id = row_number())

vertices_my <- vertex_df %>%
  mutate(id = row_number(),
         name = label,
         shortName = label,  # Assuming you don't have a shorter name
         group = ifelse(leaf, "leaf", "internal")) %>%
  select(id, name, shortName, group, leaf, height, members) %>%
  arrange(name) %>%
  mutate(name = factor(name, levels = unique(name)))

# Check for NA or empty names and replace them with a placeholder
vertices_my <- vertices_my %>%
  mutate(shortName = ifelse(is.na(shortName) | shortName == "", paste0("Node", row_number()), shortName)) %>%
  mutate(name = ifelse(is.na(name) | name == "", paste0("Node_", row_number()), name))

# Create connections_my dataframe
connections_my_df <- edge_df %>%
  left_join(vertex_df, by = c("from" = "node_id")) %>%
  left_join(vertex_df, by = c("to" = "node_id"), suffix = c("_from", "_to"))

connections_my <- connections_my_df %>%
  select(from, to) %>% #This selects only the 'from' and 'to' columns from the original dataframe.
  mutate(from = vertices_my$name[from],
         to = vertices_my$name[to]) %>% #This replaces the values in the 'from' and 'to' columns with corresponding names from the vertices_my dataframe.
  distinct() #This removes any duplicate rows from the resulting dataframe.

connections_my <- connections_my %>%
  filter(from != "" & to != "") #This keeps only the rows where from and to are not empty strings 

# Preparation to draw labels properly:
# Extract vertex data
vertices_my <- hierarchy_graph %>%
  activate(nodes) %>%
  as_tibble() %>%
  mutate(
    id = row_number(),
    name = label,
    shortName = label,  # Keep original labels
    group = ifelse(leaf, "leaf", "internal"),
    unique_name = make.unique(as.character(name), sep = "_")
  ) %>%
  select(id, name, shortName, unique_name, group, leaf, height, members)

# Correct any empty or NA names only for internal nodes
vertices_my <- vertices_my %>%
  mutate(
    name = ifelse(!leaf & (is.na(name) | name == ""), paste0("Node_", row_number()), name),
    shortName = ifelse(!leaf & (is.na(shortName) | shortName == ""), paste0("Node", row_number()), shortName)
  )

# Prepare label angles for leaves
leaf_vertices <- vertices_my %>% 
  filter(leaf) %>%
  mutate(
    leaf_id = row_number(),
    angle = 90 - 360 * (leaf_id - 1) / n(),
    hjust = ifelse(angle < -90, 1, 0),
    angle = ifelse(angle < -90, angle + 180, angle)
  )

# Update vertices_my with leaf angles
vertices_my <- vertices_my %>%
  left_join(leaf_vertices %>% select(id, angle, hjust), by = "id")

The circular dendrogram

This code plots a circular dendrogram from the hierarchical clustering results

Code

#Create the graph object 
mygraph <- graph_from_data_frame(edge_df, vertices = vertices_my)

# Basic dendrogram ----
ggraph(mygraph, layout = 'dendrogram', circular = TRUE) +
  geom_edge_link(size = 0.4, alpha = 0.1) +
  geom_node_text(aes(x = x*1.01, y = y*1.01, filter = leaf, label = shortName, angle = angle, hjust = hjust), size = 1.5, alpha = 1) +
  coord_fixed() +
  theme_void() +
  theme(
    legend.position = "none",
    plot.margin = unit(c(0,0,0,0),"cm"),
  ) +
  expand_limits(x = c(-1.2, 1.2), y = c(-1.2, 1.2))

The hierarchical edge bundling

This code plots a hierarchical edge bundling chart from the hierarchy defined in the dendrogram

Code

#Correct the connections_my dataframe with valid connections 
# Get leaf nodes
leaf_nodes <- V(mygraph)$name[V(mygraph)$leaf]

# Create all possible combinations of leaf nodes
all_combinations <- expand.grid(from = leaf_nodes, to = leaf_nodes) %>%
  mutate(
    from = as.character(from),
    to = as.character(to)
  )

# Remove self-connections and duplicate connections
connections_my_corrected <- all_combinations %>%
  filter(from != to) %>%  # Remove self-connections
  mutate(
    pair = pmin(from, to),
    pair_to = pmax(from, to)
  ) %>%
  distinct(pair, pair_to) %>%  # Remove duplicates
  select(from = pair, to = pair_to)

# Convert node names to indices
from <- match(connections_my_corrected$from, V(mygraph)$name)
to <- match(connections_my_corrected$to, V(mygraph)$name)

# Plot the hierarchical edge bundle 
ggraph(mygraph, layout = 'dendrogram', circular = TRUE) +
  geom_conn_bundle(data = get_con(from = from, to = to), 
                   alpha = 0.2,  # Reduced alpha for better visibility
                 #  colour="#69b3a2", 
                   tension = 1.5,
  aes(colour=after_stat(index))) +
  scale_edge_colour_distiller(palette = "RdPu") +
  geom_node_point(aes(filter = leaf), size = 2, color = "black") +
  geom_node_text(aes(x = x*1.05, y=y*1.05, filter = leaf, label=shortName, angle = angle, hjust=hjust), 
                 size=2, alpha=1) +
  coord_fixed() +
  theme_void() +
  theme(
    legend.position="none",
    plot.margin=unit(c(0,0,0,0),"cm"),
  ) +
  expand_limits(x = c(-1.3, 1.3), y = c(-1.3, 1.3))

With cardiometabolic biomarkers

This code plots a hierarchical edge bundling chart from the hierarchy defined in the dendrogram highlighting which markers are molecular analytes and which are cardiometabolic health markers

Code

# Step 1: Load and standardize the data
markers <- read_excel(here::here("Data/Mol_raw_data.xlsx"), sheet = 5)
markers_females=markers[,c(2:3)]

library(dplyr)

vertices_my <- vertices_my %>%
  left_join(markers_females, by = c("name" = "Females")) %>%
  rename(Markers = Type) %>%
  mutate(Markers = factor(Markers))

# Define colors for the two groups
marker_levels <- levels(vertices_my$Markers)
group_colors <- setNames(c("black", "#D2B48C"), marker_levels)

mygraph <- graph_from_data_frame(edge_df, vertices = vertices_my)

#Correct the connections_my dataframe with valid connections 
# Get leaf nodes
leaf_nodes <- V(mygraph)$name[V(mygraph)$leaf]

# Create all possible combinations of leaf nodes
all_combinations <- expand.grid(from = leaf_nodes, to = leaf_nodes) %>%
  mutate(
    from = as.character(from),
    to = as.character(to)
  )

# Remove self-connections and duplicate connections
connections_my_corrected <- all_combinations %>%
  filter(from != to) %>%  # Remove self-connections
  mutate(
    pair = pmin(from, to),
    pair_to = pmax(from, to)
  ) %>%
  distinct(pair, pair_to) %>%  # Remove duplicates
  select(from = pair, to = pair_to)

# Convert node names to indices
from <- match(connections_my_corrected$from, V(mygraph)$name)
to <- match(connections_my_corrected$to, V(mygraph)$name)

# Plot the hierarchical edge bundle with colored points
ggraph(mygraph, layout = 'dendrogram', circular = TRUE) +
  geom_conn_bundle(data = get_con(from = from, to = to), 
                   tension = 1.5,
                   aes(colour = after_stat(index)),
                   show.legend = FALSE) +  # Hide edge color legend
  scale_edge_colour_distiller(palette = "RdPu") +
  geom_node_point(aes(filter = leaf, color = Markers  # Use 'group' for coloring
                      ),
                  size = 2, alpha = 1) +
  geom_node_text(aes(x = x * 1.05, y = y * 1.05,
                     filter = leaf, label = shortName, 
                     angle = angle, hjust = hjust), 
                 size = 2, alpha = 1) +
  scale_color_manual(values = group_colors, name = NULL) +  # Name the legend
  coord_fixed() +
  theme_void() +
  theme(
    legend.position = "bottom",
    plot.margin = unit(c(0,0,0,0),"cm"),
  ) +
  expand_limits(x = c(-1.3, 1.3), y = c(-1.3, 1.3))

Nbclust

The NbClust function is being used to determine the optimal number of clusters in our data. It applies multiple clustering validity indices and methods to suggest the best number of clusters, providing a consensus view based on various criteria to help make a more robust decision about the appropriate cluster count for our dataset.

Code

library(stringr)

library(NbClust)
methods <- c("ward.D2")

for (m in methods) {
  # Capture all output
  output <- capture.output({
    result <- NbClust(data = scaled_data, 
                      distance = "euclidean", 
                      min.nc = 2, 
                      max.nc = 10, 
                      method = m, 
                      index = "all")
  })
  
  # Join the output into a single string
  output_text <- paste(output, collapse = "\n")
  
  # Split the output by the asterisk separator
  split_output <- str_split(output_text, "\\*{10,}")[[1]]
  
  # Check if we have at least 3 parts
  if (length(split_output) >= 3) {
    # Trim whitespace from the second part
    second_part <- str_trim(split_output[2])
    
    # Split the second part into lines
    lines <- str_split(second_part, "\n")[[1]]
    
    # Format the output
    formatted_output <- character(0)
    for (line in lines) {
      if (str_detect(line, "Conclusion")) {
        formatted_output <- c(formatted_output, "", "Conclusion", "")
      } else if (str_detect(line, "^\\*")) {
        formatted_output <- c(formatted_output, line)
      }
    }
    
    # Join the formatted lines
    formatted_output <- paste(formatted_output, collapse = "\n")
    
    # Print the results
    cat(paste("Results for method:", m, "\n\n"))
    cat(formatted_output)
    cat("\n\n")  # Add some space between results for different methods
  } else {
    cat(paste("Unexpected output format for method:", m, "\n"))
  }
}

Results for method: ward.D2

• Among all indices:
  
• 4 proposed 2 as the best number of clusters
• 8 proposed 3 as the best number of clusters
• 1 proposed 4 as the best number of clusters
• 2 proposed 5 as the best number of clusters
• 1 proposed 8 as the best number of clusters
• 1 proposed 9 as the best number of clusters
• 6 proposed 10 as the best number of clusters

Conclusion

• According to the majority rule, the best number of clusters is 3

Plot the clusters

Code

cluster_groups <- cutree(hc, k = 3)

# Add cluster information to vertices_my
vertices_my <- vertices_my %>%
  mutate(cluster = factor(cluster_groups[match(name, names(cluster_groups))]))

library(RColorBrewer)

# Select three colors from the "Paired" palette
cluster_colors <- brewer.pal(3, "Paired")

#Create the graph object 
mygraph <- graph_from_data_frame(edge_df, vertices = vertices_my)

#Correct the connections_my dataframe with valid connections 
# Get leaf nodes
leaf_nodes <- V(mygraph)$name[V(mygraph)$leaf]

# Create all possible combinations of leaf nodes
all_combinations <- expand.grid(from = leaf_nodes, to = leaf_nodes) %>%
  mutate(
    from = as.character(from),
    to = as.character(to)
  )

# Remove self-connections and duplicate connections
connections_my_corrected <- all_combinations %>%
  filter(from != to) %>%  # Remove self-connections
  mutate(
    pair = pmin(from, to),
    pair_to = pmax(from, to)
  ) %>%
  distinct(pair, pair_to) %>%  # Remove duplicates
  select(from = pair, to = pair_to)

# Convert node names to indices
from <- match(connections_my_corrected$from, V(mygraph)$name)
to <- match(connections_my_corrected$to, V(mygraph)$name)

# Plot the hierarchical edge bundle with colored points
ggraph(mygraph, layout = 'dendrogram', circular = TRUE) +
  geom_conn_bundle(data = get_con(from = from, to = to), 
                   tension = 1.5,
                   aes(colour=after_stat(index))) +
  scale_edge_colour_distiller(palette = "RdPu") +
  geom_node_point(aes(filter = leaf, color = cluster, 
                      x = x * 1.075, y = y * 1.075),  # Move points slightly outward
                  size = 5, alpha = 0.6) +
  geom_node_text(aes(x = x * 1.15, y = y * 1.15,  # Move text even further outward
                     filter = leaf, label = shortName, 
                     angle = angle, hjust = hjust), 
                 size = 2, alpha = 1) +
  scale_color_manual(values = cluster_colors) +
  coord_fixed() +
  theme_void() +
  theme(
    legend.position = "none",
    plot.margin = unit(c(0,0,0,0), "cm"),
  ) +
  expand_limits(x = c(-1.3, 1.3), y = c(-1.3, 1.3))  # Further expanded limits to accommodate text and points

Male data

Load data and create the dendrogram

This code loads the male data, standardizes the data to one unit variance across sampled groups, calculates euclidean distances, performs hierarchical clustering and creates a dendrogram.

Code

# Step 1: Load and standardize the data
data <- read_excel(here::here("Data/Mol_raw_data.xlsx"), sheet = 4)
data <- as.data.frame(data)

# Ensure the first column becomes the row names
rownames(data) <- data[,1]
data <- data[,-1]

#Standardise the data 
scaled_data <- vegan::decostand(data, method="standardize", MARGIN = 1)

# Step 2: Calculate Euclidean distances
dist_matrix <- dist(scaled_data, method = "euclidean")

# Step 3: Apply hierarchical clustering
hc <- hclust(dist_matrix, method = "ward.D2")

# Step 4 : Convert to dendrogram 
dendro <- as.dendrogram(hc)

Create dataframes

This code creates three dataframes necessary for plotting the charts: edges, vertices and connections

Code

#Step 5: Use ggraph 
hierarchy_graph = as_tbl_graph(dendro)

#Hierarchal dataframe 
edge_df <- hierarchy_graph %>%
  activate(edges) %>%
  as_tibble() %>%
  rename(from = from, to = to)

# Extract vertices dataframe
vertex_df <- hierarchy_graph %>%
  activate(nodes) %>%
  as_tibble() %>%
  mutate(node_id = row_number())

vertices_my <- vertex_df %>%
  mutate(id = row_number(),
         name = label,
         shortName = label,  # Assuming you don't have a shorter name
         group = ifelse(leaf, "leaf", "internal")) %>%
  select(id, name, shortName, group, leaf, height, members) %>%
  arrange(name) %>%
  mutate(name = factor(name, levels = unique(name)))

# Check for NA or empty names and replace them with a placeholder
vertices_my <- vertices_my %>%
  mutate(shortName = ifelse(is.na(shortName) | shortName == "", paste0("Node", row_number()), shortName)) %>%
  mutate(name = ifelse(is.na(name) | name == "", paste0("Node_", row_number()), name))

# Create connections_my dataframe
connections_my_df <- edge_df %>%
  left_join(vertex_df, by = c("from" = "node_id")) %>%
  left_join(vertex_df, by = c("to" = "node_id"), suffix = c("_from", "_to"))

connections_my <- connections_my_df %>%
  select(from, to) %>% #This selects only the 'from' and 'to' columns from the original dataframe.
  mutate(from = vertices_my$name[from],
         to = vertices_my$name[to]) %>% #This replaces the values in the 'from' and 'to' columns with corresponding names from the vertices_my dataframe.
  distinct() #This removes any duplicate rows from the resulting dataframe.

connections_my <- connections_my %>%
  filter(from != "" & to != "") #This keeps only the rows where from and to are not empty strings 

# Preparation to draw labels properly:
# Extract vertex data
vertices_my <- hierarchy_graph %>%
  activate(nodes) %>%
  as_tibble() %>%
  mutate(
    id = row_number(),
    name = label,
    shortName = label,  # Keep original labels
    group = ifelse(leaf, "leaf", "internal"),
    unique_name = make.unique(as.character(name), sep = "_")
  ) %>%
  select(id, name, shortName, unique_name, group, leaf, height, members)

# Correct any empty or NA names only for internal nodes
vertices_my <- vertices_my %>%
  mutate(
    name = ifelse(!leaf & (is.na(name) | name == ""), paste0("Node_", row_number()), name),
    shortName = ifelse(!leaf & (is.na(shortName) | shortName == ""), paste0("Node", row_number()), shortName)
  )

# Prepare label angles for leaves
leaf_vertices <- vertices_my %>% 
  filter(leaf) %>%
  mutate(
    leaf_id = row_number(),
    angle = 90 - 360 * (leaf_id - 1) / n(),
    hjust = ifelse(angle < -90, 1, 0),
    angle = ifelse(angle < -90, angle + 180, angle)
  )

# Update vertices_my with leaf angles
vertices_my <- vertices_my %>%
  left_join(leaf_vertices %>% select(id, angle, hjust), by = "id")

The circular dendrogram

This code plots a circular dendrogram from the hierarchical clustering results

Code

#Create the graph object 
mygraph <- graph_from_data_frame(edge_df, vertices = vertices_my)

# Basic dendrogram ----
ggraph(mygraph, layout = 'dendrogram', circular = TRUE) +
  geom_edge_link(size = 0.4, alpha = 0.1) +
  geom_node_text(aes(x = x*1.01, y = y*1.01, filter = leaf, label = shortName, angle = angle, hjust = hjust), size = 1.5, alpha = 1) +
  coord_fixed() +
  theme_void() +
  theme(
    legend.position = "none",
    plot.margin = unit(c(0,0,0,0),"cm"),
  ) +
  expand_limits(x = c(-1.2, 1.2), y = c(-1.2, 1.2))

The hierarchical edge bundling

This code plots a hierarchical edge bundling chart from the hierarchy defined in the dendrogram

Code

#Correct the connections_my dataframe with valid connections 
# Get leaf nodes
leaf_nodes <- V(mygraph)$name[V(mygraph)$leaf]

# Create all possible combinations of leaf nodes
all_combinations <- expand.grid(from = leaf_nodes, to = leaf_nodes) %>%
  mutate(
    from = as.character(from),
    to = as.character(to)
  )

# Remove self-connections and duplicate connections
connections_my_corrected <- all_combinations %>%
  filter(from != to) %>%  # Remove self-connections
  mutate(
    pair = pmin(from, to),
    pair_to = pmax(from, to)
  ) %>%
  distinct(pair, pair_to) %>%  # Remove duplicates
  select(from = pair, to = pair_to)

# Convert node names to indices
from <- match(connections_my_corrected$from, V(mygraph)$name)
to <- match(connections_my_corrected$to, V(mygraph)$name)

# Plot the hierarchical edge bundle 
ggraph(mygraph, layout = 'dendrogram', circular = TRUE) +
  geom_conn_bundle(data = get_con(from = from, to = to), 
                   alpha = 0.2,  # Reduced alpha for better visibility
                 #  colour="#69b3a2", 
                   tension = 1.5,
  aes(colour=after_stat(index))) +
  scale_edge_colour_distiller(palette = "BuPu") +
  geom_node_point(aes(filter = leaf), size = 2, color = "black") +
  geom_node_text(aes(x = x*1.05, y=y*1.05, filter = leaf, label=shortName, angle = angle, hjust=hjust), 
                 size=2, alpha=1) +
  coord_fixed() +
  theme_void() +
  theme(
    legend.position="none",
    plot.margin=unit(c(0,0,0,0),"cm"),
  ) +
  expand_limits(x = c(-1.3, 1.3), y = c(-1.3, 1.3))

With cardiometabolic biomarkers

This code plots a hierarchical edge bundling chart from the hierarchy defined in the dendrogram highlighting which markers are molecular analytes and which are cardiometabolic health markers

Code

# Step 1: Load and standardize the data
markers <- read_excel(here::here("Data/Mol_raw_data.xlsx"), sheet = 5)
markers_males=markers[,c(1:3)]

library(dplyr)

vertices_my <- vertices_my %>%
  left_join(markers_males, by = c("name" = "Males")) %>%
  rename(Markers = Type) %>%
  mutate(Markers = factor(Markers))

# Define colors for the two groups
marker_levels <- levels(vertices_my$Markers)
group_colors <- setNames(c("black", "#D2B48C"), marker_levels)

mygraph <- graph_from_data_frame(edge_df, vertices = vertices_my)

#Correct the connections_my dataframe with valid connections 
# Get leaf nodes
leaf_nodes <- V(mygraph)$name[V(mygraph)$leaf]

# Create all possible combinations of leaf nodes
all_combinations <- expand.grid(from = leaf_nodes, to = leaf_nodes) %>%
  mutate(
    from = as.character(from),
    to = as.character(to)
  )

# Remove self-connections and duplicate connections
connections_my_corrected <- all_combinations %>%
  filter(from != to) %>%  # Remove self-connections
  mutate(
    pair = pmin(from, to),
    pair_to = pmax(from, to)
  ) %>%
  distinct(pair, pair_to) %>%  # Remove duplicates
  select(from = pair, to = pair_to)

# Convert node names to indices
from <- match(connections_my_corrected$from, V(mygraph)$name)
to <- match(connections_my_corrected$to, V(mygraph)$name)

# Plot the hierarchical edge bundle with colored points
ggraph(mygraph, layout = 'dendrogram', circular = TRUE) +
  geom_conn_bundle(data = get_con(from = from, to = to), 
                   tension = 1.5,
                   aes(colour = after_stat(index)),
                   show.legend = FALSE) +  # Hide edge color legend
  scale_edge_colour_distiller(palette = "BuPu") +
  geom_node_point(aes(filter = leaf, color = Markers  # Use 'group' for coloring
                      ),
                  size = 2, alpha = 1) +
  geom_node_text(aes(x = x * 1.05, y = y * 1.05,
                     filter = leaf, label = shortName, 
                     angle = angle, hjust = hjust), 
                 size = 2, alpha = 1) +
  scale_color_manual(values = group_colors, name = NULL) +  # Name the legend
  coord_fixed() +
  theme_void() +
  theme(
    legend.position = "bottom",
    plot.margin = unit(c(0,0,0,0),"cm"),
  ) +
  expand_limits(x = c(-1.3, 1.3), y = c(-1.3, 1.3))

Nbclust

The NbClust function is being used to determine the optimal number of clusters in our data. It applies multiple clustering validity indices and methods to suggest the best number of clusters, providing a consensus view based on various criteria to help make a more robust decision about the appropriate cluster count for our dataset.

Code

library(NbClust)
library(dplyr)
library(purrr)

all_indices <- c("kl", "ch", "hartigan", "cindex", "db", "silhouette", "duda", "pseudot2", 
                 "beale", "ratkowsky", "ball", "ptbiserial", "gap", "mcclain", 
                 "gamma", "gplus", "tau", "dunn", "hubert", "sdindex", "dindex", "sdbw")

run_nbclust <- function(index) {
  tryCatch({
    # Suppress output
    result <- capture.output({
      nb_result <- NbClust(data = scaled_data, 
                           distance = "euclidean", 
                           min.nc = 2, 
                           max.nc = 10, 
                           method = "ward.D2", 
                           index = index)
    })
    return(list(index = index, result = nb_result, error = NULL))
  }, error = function(e) {
    return(list(index = index, result = NULL, error = conditionMessage(e)))
  })
}

results <- map(all_indices, run_nbclust)

Code

# Extract working indices and their results
working_results <- results %>% 
  keep(~ is.null(.$error)) %>% 
  map(~ .$result)

# Combine results
if (length(working_results) > 0) {
  combined_result <- list(
    All.index = do.call(c, map(working_results, ~ .$All.index)),
    Best.nc = do.call(rbind, map(working_results, ~ .$Best.nc)),
    Best.partition = working_results[[1]]$Best.partition
  )
  
  # Count the proposed best number of clusters
  best_nc_counts <- table(combined_result$Best.nc[, 1])
  
  # Print results in the requested format
  cat("Results for method: ward.D2\n\n")
  cat("Among all indices:\n\n")
  for (nc in sort(as.numeric(names(best_nc_counts)))) {
    count <- best_nc_counts[as.character(nc)]
    cat(sprintf("* %d proposed %d as the best number of clusters\n", count, nc))
  }
  
  cat("\nConclusion\n\n")
  best_nc <- as.numeric(names(which.max(best_nc_counts)))
  cat(sprintf("According to the majority rule, the best number of clusters is %d\n", best_nc))
} else {
  cat("No indices worked successfully.\n")
}

Results for method: ward.D2

Among all indices:

• 6 proposed 2 as the best number of clusters
• 3 proposed 3 as the best number of clusters
• 1 proposed 4 as the best number of clusters
• 2 proposed 5 as the best number of clusters
• 1 proposed 7 as the best number of clusters
• 7 proposed 10 as the best number of clusters

Conclusion

According to the majority rule, the best number of clusters is 10

Plot the clusters

Code

cluster_groups <- cutree(hc, k = 10)

# Add cluster information to vertices_my
vertices_my <- vertices_my %>%
  mutate(cluster = factor(cluster_groups[match(name, names(cluster_groups))]))

library(RColorBrewer)

# Select three colors from the "Paired" palette
cluster_colors <- brewer.pal(10, "Paired")

#Create the graph object 
mygraph <- graph_from_data_frame(edge_df, vertices = vertices_my)

#Correct the connections_my dataframe with valid connections 
# Get leaf nodes
leaf_nodes <- V(mygraph)$name[V(mygraph)$leaf]

# Create all possible combinations of leaf nodes
all_combinations <- expand.grid(from = leaf_nodes, to = leaf_nodes) %>%
  mutate(
    from = as.character(from),
    to = as.character(to)
  )

# Remove self-connections and duplicate connections
connections_my_corrected <- all_combinations %>%
  filter(from != to) %>%  # Remove self-connections
  mutate(
    pair = pmin(from, to),
    pair_to = pmax(from, to)
  ) %>%
  distinct(pair, pair_to) %>%  # Remove duplicates
  select(from = pair, to = pair_to)

# Convert node names to indices
from <- match(connections_my_corrected$from, V(mygraph)$name)
to <- match(connections_my_corrected$to, V(mygraph)$name)

# Plot the hierarchical edge bundle with colored points
ggraph(mygraph, layout = 'dendrogram', circular = TRUE) +
  geom_conn_bundle(data = get_con(from = from, to = to), 
                   tension = 1.5,
                   aes(colour=after_stat(index))) +
  scale_edge_colour_distiller(palette = "BuPu") +
  geom_node_point(aes(filter = leaf, color = cluster, 
                      x = x * 1.075, y = y * 1.075),  # Move points slightly outward
                  size = 5, alpha = 0.6) +
  geom_node_text(aes(x = x * 1.15, y = y * 1.15,  # Move text even further outward
                     filter = leaf, label = shortName, 
                     angle = angle, hjust = hjust), 
                 size = 2, alpha = 1) +
  scale_color_manual(values = cluster_colors) +
  coord_fixed() +
  theme_void() +
  theme(
    legend.position = "none",
    plot.margin = unit(c(0,0,0,0), "cm"),
  ) +
  expand_limits(x = c(-1.3, 1.3), y = c(-1.3, 1.3))  # Further expanded limits to accommodate text and points